Method and system for torque/force control of hydraulic actuators

ABSTRACT

A combined scheme of identification and torque control is provided for rotary hydraulic actuators. The composite controller consists of a dynamic feedback linearizing inner loop cascaded with a robust linear feedback outer loop. The proposed controller allows the actuator to generate desired torque irrespective of the actuator motion. In fact, the controller reduces significantly the impedance of the actuator as seen by its external load, making the system an ideal source of torque suitable for many robotics and automation applications. An identification method to extract the parameters of non-linear model of actuator dynamics and to estimate a bound for modeling uncertainty, used for synthesis of the outer optimal controller, is also presented. Results are illustrated experimentally on a pitch actuator of a Schilling industrial robot.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of priority of U.S. ProvisionalPatent Application No. 60/631,990 filed Dec. 1, 2004, which isincorporated herein by reference.

FIELD OF THE INVENTION

The present invention relates generally to control systems and theirdesign. More particularly, the present invention relates to a torquecontrol system for a hydraulic actuator and the design of such system.

BACKGROUND OF THE INVENTION

Hydraulic actuators are widely used to drive robotic manipulators inindustry for tasks such as earth moving, material handling, constructionand manufacturing automation due to their large power-to-mass ratio.However, precise control of hydraulic actuators is more difficult thancontrol of conventional electric motors due to the presence ofnon-linear flow-pressure characteristics, such as: variations in thetrapped fluid volume in each actuator chamber; fluid compressibility;friction between moving parts; variations of hydraulic parameters;presence of leakage; and transmission non-linearities.

Much of the work on hydraulic control relies on linear control designmethodology that is based on local liberalization of the actuatordynamics about a nominal operating point. However, these methods sufferfrom two major drawbacks: First, since the actuator dynamics are highlynon-linear, a single linear time invariant controller can be only tunedfor a particular operating point and the performance degrades as thesystem state moves away from the operating point. Second, since thedynamics of actuator and load are coupled together, the dynamics of theload (which can be very complex) are implicitly embedded in thelinearized model of the actuator, which complicates the treatment of thesituation; it is then difficult to achieve precise control. As a result,these methods rely highly on the knowledge of the load characteristicsand variation in those characteristics.

Mechatronics systems, such as electro-hydraulic robot manipulators, areessentially multi-dimensional non-linear systems composed of mechanicaland actuator subsystems accounting for load dynamics and actuatordynamics, respectively. The control problem can be greatly simplified inmany applications, if actuators behave as an ideal source offorce/torque with low impedance, i.e. similar to electric motors.However, the force/torque generated by a hydraulic actuator is affectedby its own motion resulting in a coupled dynamics of the actuator andload.

To account for parametric uncertainty of estimated and/or somewhatinaccurate parameters, non-linear adaptive control methods havepreviously been employed. An adaptive robust method can also be used,which takes the nonparametric unmodeled dynamics into account byassuming a known bound on the nonparametric uncertainty.

Dynamic feedback linearization has been used to attempt to cancel outthe actuator non-linear dynamics. The advantage of this method is thatfor force control purpose, no knowledge of load dynamics is requiredbecause it cancels out the effect of velocity perturbation. However, inpractice, exact cancellation of the actuator dynamics is not possibledue to parametric and nonparametric modeling uncertainties. This problemhas been addressed by transforming the linearized system into standardlinear fractional uncertain structures; however, no method has beenpresented for computation of uncertainty bounds.

The control of a torque/force output is very different in nature fromknown attempts to control motion or position. When a controller is forcontrolling a system in which there is no motion, or negligible motion,much of the hydraulic behaviour is masked, and friction is the mainobservable factor affecting the system. This is one reason why manyknown systems seek to compensate for frictional components. It is alsonecessary to consider the dynamics of the whole system together, namelythe combination of the actuator and the load. In torque/force controlsituations such as those discussed according to embodiments of thepresent invention, it is necessary to consider the effect of velocity onthe system; as such, the torque/force control problem is quite differentin nature from the motion and displacement control problems.

It is, therefore, desirable to provide a procedure for identification ofactuator non-linear dynamics and quantification of modelling error. Mostexisting adaptive methods deal only with parametric uncertainties andsome robust adaptive schemes assume a known bound for non-parametricuncertainties in actuator non-linear dynamics. No method is known toestimate this bound, and attempts to account for non-linear dynamicshave drawbacks.

SUMMARY OF THE INVENTION

It is an object of the present invention to obviate or mitigate at leastone disadvantage of previous torque/force controllers and methods oftheir design.

The robot control problem is simplified by minimizing the couplingbetween the two sub-systems that can be achieved by minimizing theeffect of velocity disturbance on actuator torque. Then the robotcontrol problem is effectively reduced to the torque control of thehydraulic actuator and the control of the multi-body dynamics of amanipulator that traditionally relies on torque control inputs.

In a first aspect, the present invention provides a controller for ahydraulic actuator, the hydraulic actuator being for generating amanipulating influence to be applied to a load. The controller includesa linearizing controller for storing a linear model representingnon-linear dynamic behaviour of an unloaded hydraulic actuator. Thelinearizing controller is shown as part of an inner loop. A robustlinear controller is also provided for compensating for non-linearitiesin the linear model using an uncertainty model having a non-linearcomponent, and an estimated bound for the uncertainty model. The robustlinear controller is shown as part of an outer loop, preferably cascadedwith the inner loop.

In the case where the hydraulic actuator includes a joint, thenon-linear dynamic behaviour of the unloaded hydraulic actuator can beobtained by substantially minimizing effects of the load on themanipulating influence. This is achieved by perturbing the linear modelin response to a velocity of the joint. The linear model can be based onmeasured linear parameters of the hydraulic actuator. The linearizingcontroller can include means for obtaining the linear model based on alinearizing control law for the hydraulic actuator, or means fordetermining the linearizing control law for the hydraulic actuator. Thecontroller can further include means for calculating the estimatedbound. The manipulating influence can include a torque or a force, andthe hydraulic actuator can be a rotary hydraulic actuator or a linearhydraulic actuator.

In another aspect, the present invention provides a control architecturefor torque control of a hydraulic actuator. The control architectureincludes a dynamic feedback linearizing inner loop, and a robust linearfeedback outer loop. The outer loop is cascaded with the inner loop topermit the actuator to generate a desired torque irrespective of motionof the hydraulic actuator.

In a further aspect, the present invention provides a method ofdesigning a hydraulic actuator controller. The method includes thefollowing steps: determining an uncertainty model for a linearized modelof the hydraulic actuator, the uncertainty model including a non-linearcomponent; estimating an uncertainty bound, for the uncertainty model,based on identified parameters of non-linear behaviour of the actuator;and designing a robust linear controller based on the determineduncertainty model and the estimated uncertainty bound.

The step of estimating the uncertainty bound can include the followingsteps: applying a linearizing control law using control input as anexcitation signal; identifying a linear discrete time model as theuncertainty model, based on measured values of torque and control input;and computing a minimum value of the uncertainty bound such that theuncertainty model is not invalidated by the measured values of torqueand control input.

The non-linear component can be based on unmodeled actuator dynamics.The linearizing control law can be a dynamic feedback linearizingcontrol law. The step of designing the robust linear controller caninclude imposing robust stability and performance constraints based oncharacteristics of the uncertainty model. The method can further includethe step of calculating a linearizing control law based on theidentified parameters of non-linear behaviour of the actuator. Themethod can further include the step of extracting the identifiedparameters based on measured signals.

According to another aspect, the present invention provides acomputer-readable storage medium, comprising statements and instructionswhich, when executed, cause a computer to perform a method according toembodiments of the present invention.

Other aspects and features of the present invention will become apparentto those ordinarily skilled in the art upon review of the followingdescription of specific embodiments of the invention in conjunction withthe accompanying figures.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the present invention will now be described, by way ofexample only, with reference to the attached Figures, wherein:

FIG. 1A is a block diagram of a force controller according to anembodiment of the present invention;

FIG. 1B is a block diagram of a torque controller according to anembodiment of the present invention;

FIG. 2 is a flowchart illustrating a method of designing a hydraulicactuator controller according to an embodiment of the present invention;

FIG. 3 is a block diagram of an uncertainty model structure according toan embodiment of the present invention;

FIG. 4 is a block diagram of an uncertainty model structure according toanother embodiment of the present invention;

FIG. 5 graphically illustrates a validation of a non-linear modelaccording to an embodiment of the present invention;

FIG. 6 graphically illustrates a frequency response of a linearizedmodel represented by an uncertainty model structure according to anembodiment of the present invention;

FIG. 7 graphically illustrates a limit cycle in actuator dynamics in thepresence of a high gain external controller;

FIG. 8 graphically illustrates frequency responses of resulting inputand output sensitivity functions together with weighing functions;

FIG. 9 graphically illustrates an effect of external torque disturbanceon generated torque;

FIG. 10 graphically illustrates step tracking of actuator torque in thepresence of velocity;

FIG. 11 graphically illustrates 0.5 Hz sine tracking of actuator torquein presence of velocity;

FIG. 12 graphically illustrates 2 Hz sine tracking of actuator torque inpresence of velocity;

FIG. 13 graphically illustrates 5 Hz sine tracking of actuator torque inpresence of velocity;

FIG. 14 graphically illustrates step response of actuator torque inabsence of velocity;

FIG. 15 graphically illustrates 0.5 Hz sine tracking of actuator torquein absence of velocity;

FIG. 16 graphically illustrates 2 Hz sine tracking of actuator torque inabsence of velocity; and

FIG. 17 graphically illustrates 5 Hz sine tracking of actuator torque inabsence of velocity.

DETAILED DESCRIPTION

Generally, the present invention provides a method and system foridentification and torque/force control for hydraulic actuators, such asrotary hydraulic actuators. The methodology can be readily applied tolinear hydraulic actuators. The composite controller consists of adynamic feedback linearizing inner loop cascaded with a robust linearfeedback outer loop. The proposed controller allows the actuator togenerate desired torque irrespective of the actuator motion. In fact,the controller reduces significantly the impedance of the actuator asseen by its external load, making the system an ideal, or substantiallyideal, source of torque suitable for many robotics and automationapplications. An identification method to extract the parameters ofnon-linear model of actuator dynamics and to estimate a bound formodeling uncertainty, used for synthesis of the outer optimalcontroller, is also presented. Results are illustrated experimentally ona pitch actuator of a Schilling industrial robot.

In the realm of hydraulic actuators, rotary hydraulic actuators producea torque output, whereas linear hydraulic actuators produce a forceoutput. The term “manipulating influence” is used herein to representeither a torque or a force, and is used as a generic term to cover bothpossibilities, depending on whether a hydraulic actuator is linear orrotary. The terms “torque/force control” and “hydraulic actuator outputcontrol” are used herein to refer to the control of the output of thehydraulic actuator, whether it is a linear or rotary actuator. The terms“torque/force controller” and “hydraulic actuator output controller” asused herein represent a controller for controlling force (in the case ofa rotary actuator) and/or for controlling torque (in the case of alinear actuator). Of course, the term “controller” is also used hereinto refer to a torque/force controller. The term “velocity” is usedherein to represent a speed and trajectory (either linear or angular).Though reference is made to actuator velocity, joint velocity, and loadvelocity, it is to be understood that each of these velocities isdescribing the same velocity in the case where an actuator is applying atorque/force to a load. As such, a calculation of load velocity can beused to determine actuator velocity.

The control of a torque/force output is very different in nature fromknown attempts to control motion or position. When a controller is forcontrolling a system in which there is no motion, or negligible motion,much of the hydraulic behaviour is masked, and friction is the mainobservable factor affecting the system. This is one reason why manyknown systems seek to compensate for frictional components. It is alsonecessary to consider the dynamics of the whole system together, namelythe combination of the actuator and the load. In torque/force controlsituations such as those discussed according to embodiments of thepresent invention, it is necessary to consider the effect of velocity onthe system; as such, the torque/force control problem is quite differentin nature from the motion and displacement control problems. Intorque/force control according to embodiments of the present invention,the two subsystems are decoupled by way of velocity feedback. Thisresults in system modularity. It is no longer required to model theenvironment, since the actuator can be controlled independently of theeffects of the load.

The robot control problem is greatly simplified by minimizing thecoupling between the two sub-systems that can be achieved by minimizingor eliminating the effect of velocity disturbance on actuator torque.Then the robot control problem is reduced to the torque control ofhydraulic actuator and the control of the multi-body dynamics ofmanipulator that traditionally relies on torque control inputs.

A combined scheme of identification and torque control is described forhydraulic actuators, such as rotary hydraulic actuators. The methodologyis readily applicable for linear hydraulic actuators. The compositecontroller includes a dynamic feedback linearizing inner loop cascadedwith an optimal robust linear feedback outer loop. The proposedcontroller allows the actuator to generate desired torque irrespectiveof the actuator motion. In fact, the controller reduces significantlythe impedance of the actuator as seen by its external load, making thesystem an ideal source of torque suitable for many robotics andautomation applications. Discussion of and “ideal” source herein is tobe understood as referring to a source having behaviour that issubstantially ideal.

The controller allows a hydraulic actuator to generate desired torque orforce regardless of actuator motion. It can reduce significantly theimpedance of the actuator as seen by its external load, making thesystem an ideal, or near ideal, source of torque or force suitable formany robotics and automation applications. The dynamic feedbacklinearizing controller can be constructed based on an identificationprocedure which identifies parameters of actuator non-linear model.Since the feedback-linearized model is not perfectly “linear”, a novelidentification procedure is developed to fit an “uncertain” modelstructure to the almost linearized system. The robust outer-loopcontroller permits to consider different types of performance objective,either in time domain or in frequency domain. The proposed linear robustouter-loop controller presents a very efficient means for attenuatinglimit-cycle oscillations in servo-valve dynamics, and is discrete andeasily implementable. All design procedures can be realized by means ofpowerful convex optimization algorithms.

The identification and control scheme according to embodiments of thepresent invention relies neither on a priori knowledge of load dynamicsnor external torques, and makes the controlled actuator as a source oftorque with low impedance, i.e. acting virtually as an electro-motor.Moreover, no a prior assumption is made on a bound for actuatorunmodeled dynamics. The bound is estimated via an identificationprocedure. The controller synthesis procedure allows imposing severalperformance objectives either in time-domain or frequency domain and itcomprises numerically tractable convex optimizations. An identificationmethod to extract the parameters of non-linear model of actuatordynamics and to estimate a bound for modelling uncertainty, used forsynthesis of the outer optimal controller, is also presented.

FIG. 1A is a block diagram of a force controller according to anembodiment of the present invention. This figure is similar in nature toFIG. 1B, except that it illustrates the system more generally, andindicates forces rather than torques. As mentioned earlier, it is thetype of actuator (linear or rotary) that determines whether embodimentsof the present invention will act to control force or to control torque;the system itself is the same in either case. The controller is for ahydraulic actuator, the hydraulic actuator being for generating amanipulating influence to be applied to a load. The controller includesa linearizing controller for storing a linear model representingnon-linear dynamic behaviour of an unloaded hydraulic actuator. Thelinearizing controller is shown as part of an inner loop. A robustlinear controller is also provided for compensating for non-linearitiesin the linear model using an uncertainty model having a non-linearcomponent, and an estimated bound for the uncertainty model. The robustlinear controller is shown as part of an outer loop, preferably cascadedwith the inner loop.

In the case where the hydraulic actuator includes a joint, thenon-linear dynamic behaviour of the unloaded hydraulic actuator can beobtained by substantially minimizing effects of the load on themanipulating influence. This is achieved by perturbing the linear modelin response to a velocity of the joint. The linear model can be based onmeasured linear parameters of the hydraulic actuator. The linearizingcontroller can include means for obtaining the linear model based on alinearizing control law for the hydraulic actuator, or means fordetermining the linearizing control law for the hydraulic actuator. Thecontroller can further include means for calculating the estimatedbound. The manipulating influence can include a torque or a force, andthe hydraulic actuator can be a rotary hydraulic actuator or a linearhydraulic actuator.

Described in a different manner, an embodiment of the present inventionprovides a control architecture for torque control of a hydraulicactuator. The control architecture includes a dynamic feedbacklinearizing inner loop, and a robust linear feedback outer loop. Theouter loop is cascaded with the inner loop to permit the actuator togenerate a desired torque irrespective of motion of the hydraulicactuator.

As shown in FIG. 1B, the actuator velocity and position are produced asa result of actuator torque τ superimposed with external torque τ_(ext).On the other hand, actuator velocity and motion, in turn, affectactuator torque through non-linear actuator dynamics. In thisapplication the effect of velocity on generated torque is considered asdisturbance. As such, the control objective is to solve two tasks:compensate for non-linear dynamics for achieving accurate referencetracking; and minimize sensitivity to velocity disturbance, that isequivalent to increase the back-drivability of the actuator or todecrease its impedance.

According to an embodiment of the present invention, the proposedcontroller scheme includes three main stages, with an optional initialstage, which can be implemented first. In the initial stage, parametersof the actuator non-linear model are identified. This identification canbe achieved using the measured signals (actuator position, velocity, andchamber pressures) and by a standard least squares algorithm.

In the first main stage, using the identified parameters of the actuatornon-linear model, a dynamic feedback linearizing control law u* iscalculated (See FIG. 1B). The term “control law” as used in thisspecification is a common term in the art used to designate a controlrelationship that has been determined to be true for a particulararrangement. Neglecting the servo-valve dynamics, this command ideallytransforms the non-linear system into a simple integrator if ν isconsidered as a new input and τ as output. However, in the course ofidentification experiments it becomes obvious that there arenon-negligible discrepancies between the non-linear model of actuatorwhose parameters are optimally identified and the true system. Thisimplies that the non-linear model cannot perfectly capture the dynamicalbehavior of the actuator and that the dynamic mapping from ν to τdiffers from an integrator. Therefore, having implemented the feedbacklinearization, we perform the second round of identification to fit anuncertainty model structure to the dynamic mapping from ν to τ. Thisuncertainty model structure includes a linear time-invariant (LTI) modelĜ and a non-linear operator Δ that represents both the parametric andnon-parametric uncertainties of the actuator model (F(Ĝ, Δ) in FIG. 1B).Herein no assumption is made on the value of the uncertainty bound;instead, this bound is estimated from the identification procedure. Itis worthwhile to notice that the model uncertainty represented by theoperator Δ is non-linear and the standard methods for uncertaintybounding in system identification theory are not applicable in thiscase. The proposed identification method is based on the recent resultsin model validation.

The third stage of the proposed method is to design the external linearcontroller C satisfying several performance and robust stabilityrequirements. Specifically, we translate these requirements into I₁ orH_(∞) control design specifications that in turn, are formulated by somemixed Linear Programming and Linear Matrix Inequality constraints.

Based on the identified non-linear model, a dynamic feedback linearizingcontrol law can be calculated. This command should ideally transform thenon-linear system into a simple Integrator. However, due to imperfectparameter identification and the presence of unmodeled dynamics, thefeedback-linearized system may deviate from a simple integrator model.

To describe the foregoing in a different manner, reference is made toFIG. 2, which is a flowchart illustrating a method of designing ahydraulic actuator controller according to an embodiment of the presentinvention. The method of designing a hydraulic actuator controlleraccording to an embodiment of the present invention includes thefollowing steps: determining an uncertainty model for a linearized modelof the hydraulic actuator, the uncertainty model including a non-linearcomponent; estimating an uncertainty bound, for the uncertainty model,based on identified parameters of non-linear behaviour of the actuator;and designing a robust linear controller based on the determineduncertainty model and the estimated uncertainty bound.

FIG. 2 also illustrates some optional steps, which can be included inthe step of estimating the uncertainty bound. The optional steps are asfollows: applying a linearizing control law using control input as anexcitation signal; identifying a linear discrete time model as theuncertainty model, based on measured values of torque and control input;and computing a minimum value of the uncertainty bound such that theuncertainty model is not invalidated by the measured values of torqueand control input.

The non-linear component can be based on unmodeled actuator dynamics.The linearizing control law can be a dynamic feedback linearizingcontrol law. The step of designing the robust linear controller caninclude imposing robust stability and performance constraints based oncharacteristics of the uncertainty model (such constraints will bedescribed later). The method can further include the step of calculatinga linearizing control law based on the identified parameters ofnon-linear behaviour of the actuator. The method can further include thestep of extracting the identified parameters based on measured signals.

Expressed in a slightly different manner, according to an embodiment ofthe invention, there is provided a method of designing a hydraulicactuator controller, comprising: determining an uncertainty model tocompensate for differences between a linearized model of the hydraulicactuator and actual behaviour of the hydraulic actuator, the uncertaintymodel including a non-linear component; estimating an uncertainty bound,for the uncertainty model, based on identified parameters of non-linearbehaviour of the actuator; and designing a robust linear controllerbased on the determined uncertainty model and the estimated uncertaintybound. The uncertainty model can include a linear time invariant modelcomponent.

A method according to an embodiment of the present invention canalternatively be described as an identification method for designing ahydraulic actuator controller. The method includes the following steps:determining the parameters of the non-linear model of the actuatordynamics; designing a linearizing control law based on the determinednon-linear model; fitting an “uncertain” model structure to the almostlinearized system, and estimating an uncertainty bound for the modellinguncertainty; and designing a robust linear controller based on thedetermined uncertainty model and the estimated uncertainty bound.

The model structure can include a linear time-invariant (LTI) model andan uncertainty block representing all factors that affect linearizationquality. The LTI model together with an upper-bound for the uncertaintyblock can be estimated by an identification procedure. Theidentification method can handle non-linear uncertainty blocks, asopposed to existing methods that handle mainly linear uncertainties. Theidentification scheme requires no “a priori” information on the systemdynamics.

The step of estimating the uncertainty bound can include the followingsteps: using a linearizing control input as an excitation signal andtorque as the output signal; identifying a linear discrete time model asthe uncertainty model, based on measured values of input/output signals;and computing a minimum value of the uncertainty bound such that theuncertainty model is not invalidated by the measured values of torqueand control input signals.

Actuator Dynamics

The dynamics of hydraulic pressure of the chambers assuming compressiblefluid are described by

$\begin{matrix}{{\frac{1}{\beta}\begin{bmatrix}{\overset{.}{p}1} \\{\overset{.}{p}2}\end{bmatrix}} = {{\begin{bmatrix}\frac{{- c_{l}} - c_{el}}{V_{1}(x)} & \frac{c_{l}}{V_{1}(x)} \\\frac{c_{l}}{V_{2}(x)} & \frac{{- c_{l}} - c_{el}}{V_{2}(x)}\end{bmatrix}\begin{bmatrix}{p\; 1} \\{p\; 2}\end{bmatrix}} + {p_{a}\begin{bmatrix}\frac{c_{el}}{V_{1}(x)} \\\frac{c_{el}}{V_{2}(x)}\end{bmatrix}} + {\begin{bmatrix}\frac{- D}{V_{1}(x)} \\\frac{D}{V_{2}(x)}\end{bmatrix}\omega} + {{c_{p}\begin{bmatrix}\frac{\sqrt{K_{1}}}{V_{1}(x)} \\\frac{\sqrt[ - ]{K_{2}}}{V_{2}(x)}\end{bmatrix}}\mspace{11mu} u}}} & (1)\end{matrix}$where β is the effective bulk modulus, p₁, p₂ are pressures inside thetwo chambers of actuator, x is the position angle,V₁(x)=V₀+Vx(V₂(x)=V₀−Dx) is the trapped fluid volume in the first(second) chamber, respectively. D is the volume displacement of actuatorand x∈(−D⁻¹V₀, D⁻¹V₀).

The coefficients of the internal and external leakages are denoted by c₁and c_(el), respectively. u is the spool-valve displacement andK _(i)=0.5(p _(s) −p _(a))−(−1)^(i) sgn(u)[0.5(p _(s) +p _(a))−p _(i)],i=1,2,  (2)where p_(s) is the supply pressure, p_(a) is the external pressure andc_(p) is the discharge coefficient of the valve. In this embodiment, weassume an identical discharge coefficient c_(p), for both inlet andoutlet ports of the valve, although some servovalves have larger c_(p),for the outlet ports than the inlet ports. Generally, c_(p), depends onliquid density, however in this embodiment c_(p) is considered constant.Obviously, it is possible to change the equations if differentassumptions are made.

In this embodiment we neglect the servo-valve dynamics and hence theservo-valve displacement u is treated directly as control input signal.The torque generated by a rotary hydraulic actuator τ is proportional tothe pressure difference between the two chambers, i.e.τ=D(p ₁ −p ₂)  (3)

β Effective bulk modulus D Volume displacement c_(p) Dischargecoefficient of valve c₁ Internal leakage c_(el) External leakage V_(1,2)Trapped fluid volume in chambers V₀ Initial fluid volume x Positionangle ω Angular velocity q_(1,2) Supplied flows p_(1,2) Pressure insidethe two u Servo-valve displacement chambers (control input) P_(a)External pressure τ Hydraulic torque P_(s) Supply pressure v Controlinput for linearized τ_(ref) Torque reference signal system τ_(ext)External torque disturbance γ Upperbound for uncertainty C Externalcontroller θ Vector of parameters

Input-to-torque Exact Linearization

Differentiating the actuator torque in (3) with respect to time andreplacing {dot over (p)}1 and {dot over (p)}2 from the actuator dynamicsequations yields (4) as follows:{dot over (τ)}=−βD(c ₁+c_(el))P(x)(p ₁ −p ₂)+βDc _(el) P ₁(x)p _(a) −βD² P(x)ω+βDc _(p) Q(p ₁ ,p ₂ ,x,u)uwhere ω={dot over (χ)} is the angular velocity of the actuator, andP(x), P₁(x) and Q(p₁, p₂, x, u) are defined byP(x)=V ₁(x)⁻¹ +V ₂(x)⁻¹  (5)P(x)=V ₁(x)⁻¹ −V ₂(x)⁻¹  (6)Q(p ₁ ,p ₂ ,x,u)=V ₁(x)⁻¹√{square root over (K ₁)}+V₂(x)⁻¹√{square rootover (K ₂)}  (7)

Equation (1) describes the second order dynamics of the actuator. Thefact that the command signal u appears in the first derivative of thegenerated torque shows that the relative degree of the system is one. Itis evident from Equation (4) that the actuator torque depends on twoinputs: motion variables i.e. position and velocity [x, ω] andspool-valve displacement u. Herein, the former is treated as knowndisturbance, while the latter is considered as control input. The goalof an ideal torque controller design is hence to perform precise torquetracking regardless of actuator motion.

From Equation (4), the linearizing command can be computed by

$\begin{matrix}{u^{*} = \frac{\begin{matrix}{{\beta\; D\mspace{11mu}\left( {c_{l} + c_{el}} \right)\left( {p_{1} - p_{2}} \right)\mspace{11mu} P\mspace{11mu}(x)} -} \\{{\beta\;{Dc}_{el}p_{a}{P_{1}(x)}} + {D^{2}\beta\; P\mspace{11mu}(x)\mspace{11mu}\omega} + v}\end{matrix}}{{\beta\;{Dc}_{p}Q\mspace{11mu}\left( {p_{1},p_{2},x,{{sgn}\mspace{11mu}\left( u^{*} \right)}} \right)}\;}} & (8)\end{matrix}$where ν is the new command signal. Obviously, this control lawtransforms ν-τ map into an integrator, i.e. τ=ν. In order to implementthe linearizing command law in (8) we need to express the control signalu* explicitly in terms of the measured signals p₁, p₂, x, ω and the newinput signal ν. However, Equation (8) does not express u* in an explicitform because u* appears in the right-hand side (RHS) of (8). Thisproblem can be easily solved by observing the definitions of Q andK_(i). In fact, one can infer from (2) and (7) that Q(.) depends only onthe sign of u*

Therefore, by virtue of (8) and noting that scalars Q(.), c_(p), and βare all positive valued, we can saysgn(u*)=sgn(β(c ₁ +c _(el))(p ₁ −p ₂)P(x)−βDc _(elpa) P ₁(x)+D ²βP(x)ω+ν)which shows that u* depends only on p₁, p₂, x, ω and ν. Note that thelinearizing command (8) is applicable when Q≠0. For (p₁, p₂)=(p_(s),p_(a)) or (p₁, p₂)=(p_(a), p_(s)), Q is zero and the actuator dynamicsbecomes uncontrollable from the input. The variations of p₁ and p₂ inthis case, depend only on velocity.

Identification of Actuator Dynamics and Uncertainty Bounding

In this section we describe a two-stage procedure for parametricidentification of actuator non-linear dynamics and for quantization ofmodeling uncertainty in l₁ topology. At the first stage of thisprocedure, the parameters of the non-linear model of the actuator areidentified. It is assumed that the measurements of the pressure signalsp₁, p₂, the velocity ω, the input signal u and the position x areavailable; and the derivative of the torque signal {dot over (τ)} iscomputed by numerical differentiation. DefineY(p₁p₂,x,ω,u)=[−(p₁−p₂)P(x),−Dp_(a)P₁(x),−D²P(x)ω, DQ(.)u] andθ=[β(c₁+c_(el)), βc_(el),β,βc_(p)]^(T), then equation (4) can beexpressed in standard linear regression form{dot over (τ)}=Y(p ₁ ,p ₂ ,x,ω,u)θ  (9)

The estimated parameter vector {circumflex over (θ)} is the solution tothe following convex optimization problem

$\begin{matrix}{\min\limits_{\hat{\theta}}{{\overset{.}{\tau} - {Y\mspace{11mu}\left( {p_{1},p_{2},x,\omega,u} \right)\mspace{11mu}\hat{\theta}}}}_{p}} & (10)\end{matrix}$where ∥.∥_(p) denotes signal p-norm.

However, in practice the noise caused by numerical differentiation of τis not negligible. In order to analyze the effect of noise on theidentification problem (10), let us denote e_({dot over (τ)}) as thenoise introduced by numerical differentiation of τ and define

$ɛ = {\frac{{e_{\overset{.}{\tau}}}_{2}}{{\overset{.}{\tau}}_{2}}.}$Also, define γ=[Y(t_(o))^(T), Y(t₁)^(T), . . . ,Y(t_(N))^(T)]^(T). Letκ≧1 be the condition number of Y. Obviously, a large κ indicates thatthe regression matrix Y is close to singularity. If (10) is consideredas a least squares problem (p=2), then it can be shown that for ε<k−1,we have∥Y{tilde over (θ)}∥₂≦2(1+κ)∥e _({dot over (τ)})∥₂+∥{dot over (τ)}∥₂O(ε²)  (11)where {tilde over (θ)}=θ−{circumflex over (θ)}.

It is observed that the operator Δ enters as an additive uncertainty tothe integrator system. Now, the main problem is to compute an upperboundfor this operator. The proposed method consists of a new identificationprocedure as follows: Based on the estimated parameters {circumflex over(θ)}, the linearizing control law (8) is applied while input ν isconsidered as the excitation signal. Then, by using the measured valuesof τ and ν, we identify a linear discrete time model Ĝ and compute theminimum value of γ where ∥Δ∥₁≦γ, such that the following uncertaintymodel structureτ=(Ĝ+W Δ)ν+ e  (14)is not in validated by the experimental data τ and ν.

It is evident from this inequality that the propagation of noise to theidentification problem can be minimized when κ is close to one. Since Yis a function of input signal u, then κ obviously depends on the choiceof u. This fact suggests that the input signal has an important role inachieving the minimum possible parametric error {tilde over (θ)}. Fromexperimental point of view, if a set of feasible input signals isavailable for identification purpose, then one can choose the best inputsignal that minimizes κ

Unmodeled dynamics: On the other hand, there is always a part of theactuator dynamics that are not captured by the actuator torque dynamicsequation (4); this part is referred to as unmodeled dynamics. Accordingto an embodiment of the present invention, we represent the unmodeleddynamics by a perturbation signal d(p₁,p₂,x,ω,u,t). The unmodeleddynamics can be due to the servo-valve dynamics, hysteresis in theelectromagnetic circuit that derives the valve operation, deadband incontrol valve, delay in the servo-valve, etc. Moreover, the actuator canbe affected by any perturbation that is not a function of actuatorstates. As shown in FIG. 1, the external torque τ_(ext) is suchperturbation that affects actuator dynamics through the velocity.Therefore, the actuator torque dynamics is indeed in the form of{dot over (τ)}=Y(p ₁ ,p ₂ ,x,ω,u)θ+d(p ₁ ,p ₂ ,x,ω,u,t)which implies that if the identified parameters are used in thelinearizing command law (8), the resulting dynamics will take the form{dot over (τ)}−ν=Y(p ₁ ,p ₂ ,x,ω,u){tilde over (θ)}+d(p ₁ ,p ₂,x,ω,u,t)  (12)

The first term in the RHS of (12) refers to the parametric uncertainty.Note that the perturbation d(p₁, p₂,x,ω,u,t) is only a function of theestimated parameters (through its dependence on u) and not a function ofthe parametric error {tilde over (θ)}. As a result, we consider d(.) asthe non-parametric uncertainty. In the sequel we present two methods forrepresentation of uncertainty. Clearly, there are many otherrepresentations that can deal with specific cases of parametric ornonparametric uncertainties. The choice of each representation dependson the nature of uncertainty as well as the available tools for solvingthe resulting identification problem. In general, uncertainty can berepresented by linear fractional forms. However, the solution of theresulting model validation problem usually leads to non-convexoptimizations that are not numerically tractable (e.g. when Δ∈

H_(∞) one should solve a so-called μ problem)

Case A

FIG. 3 is a block diagram of an uncertainty model structure according toan embodiment of the present invention, described as Case A. Inparticular, FIG. 3 illustrates an additive model structure, where Ĝ is amodel for an integrator.

Assume that both the parametric and non-parametric uncertainty terms arebounded and that they can be represented by

${{Y\mspace{11mu}\left( {p_{1},p_{2},x,\omega,u} \right)\mspace{11mu}\overset{\sim}{\theta}} + {d\mspace{11mu}\left( {p_{1},p_{2},x,\omega,u,t} \right)}} = {\frac{\mathbb{d}}{\mathbb{d}t}\left( {{\Delta\; v} + {e\mspace{11mu}(t)}} \right)}$where Δ is a bounded non-linear operator with ∥Δ∥₁≦γ where the I₁ normof an operator like Δ mapping the signal x to the signal y, is definedby

${\Delta }_{1} = {\sup_{x \neq o}\frac{{y}_{\infty}}{{x}_{\infty}}}$with ∥x∥_(∞)=sup_(t)|x(t)|. Moreover, e (t) represents any perturbationthat is not a function of system states. Consequently, equation (12)becomesτ=(D ⁻¹+Δ)ν+e(t)  (13)where D⁻¹ is the integration operator.

It is observed that the operator Δ enters as an additive uncertainty tothe integrator system. Now, the main problem is to compute an upperboundfor this operator. The proposed method consists of a new identificationprocedure as follows: Based on the estimated parameters {circumflex over(θ)}, the linearizing control law (8) is applied while input ν isconsidered as the excitation signal. Then,

* by using the measured values of τ and ν, we identify a linear discretetime model Ĝ and compute the minimum value of γ where ∥Δ∥₁≦γ, such thatthe following uncertainty model structureτ=(Ĝ+WΔ)ν+e  (14)is not invalidated by the experimental data τ and ν.

Here e represents sensor noise or any other external disturbance to theactuator dynamics that is independent of system states. This signal isassumed to be bounded by∥e∥_(∞)≦σ  (15)

In literature, this type of identification scheme has been recognized asmodel validation-based identification and it is based on the modelvalidation concept. In the uncertainty model structure (14), W is aknown weighting transfer function and the model Ĝ represents the effectof integrator term in (13).

The main reason for using the l₁ norm (or the induced l_(∞) to l_(∞)norm) for characterization of the uncertainty, is due to the fact thatthe uncertainty in the actuator dynamics has non-linear characteristic.Therefore, unlike many existing methods for bounding LTI uncertaintiesin H₂ or H_(∞) topologies, herein we need to use an induced operatornorm for characterizing the non-linear model uncertainty. Moreover, theadvantage of using l₁ norm over the other induced norms is that theresulting identification problem can be solved by a linear programming.

For model G in form of a rational transfer functionĜ(q⁻¹)=B(q⁻¹)/A(q⁻¹), where q⁻¹ is a unit delay operator, the formulatedidentification problem is tantamount to solving a non-convexoptimization problem. An iterative algorithm known in the art can thenbe used to solve the problem. Here it is assumed that the model Ĝ isexpressed in terms of the orthonormal basis functions as

$\begin{matrix}{{\hat{G}\mspace{11mu}\left( q^{- 1} \right)} = {\sum\limits_{k = 0}^{n}\;{{\hat{l}}_{k}{F_{k}\left( q^{- 1} \right)}}}} & (16)\end{matrix}$where F_(k)(q⁻¹) is the k-th known orthonormal basis and {circumflexover (l)}_(k)s are the parameters to be identified.

With this model description, the parameters appear linearly and theresulting optimization will be convex. The ongoing analysis shows thatthe stated identification problem can be solved via linear programming.Let {circumflex over (l)}=[{circumflex over (l)}₀, . . . ,{circumflexover (l)}_(n)]^(T) be the vector of parameters and T_(τ) represent thefirst n columns of a lower triangular Toeplitz matrix constructed from[τ(0), . . . , τ(N)^(T)]. Moreover, F is a (N+1)×n matrix whose k−thcolumn is the first N+1 samples of the impulse response of the basisfunction F_(k). Similarly, T_(W) is a (N−1)×(N−1) lower triangularToeplitz matrix that is constructed from the first samples of theimpulse response of the known weighting function W.

Proposition 1 Suppose that N+1 samples of the experimental data ν and τare available (ν_(N) and τ_(N)) and a bound on the noise signal e as(15) is available. Then the following linear programming problemidentifies the parameters of the model Ĝ and computes the value of thesmallest γ with ∥Δ∥₁≦γ such that the model structure (1.4) is notinvalidated by the given experimental data:

$\begin{matrix}{{subject}\mspace{14mu}{to}\text{:}\begin{matrix}{\min\;\gamma} \\{\hat{l},e_{N},\gamma} \\{{{T_{W}^{- 1}\left( {\tau_{N} - e_{N} - {T_{v}\overset{\_}{F}\hat{l}}} \right)}} \leq {\gamma\; E_{v}}}\end{matrix}} & (17) \\{{e_{N}} \leq {\sigma\underset{\_}{1}}} & (18)\end{matrix}$

In the above optimization problem, the parameter vector {circumflex over(l)}, the noise vector e_(N) and the scalar γ are the optimizationvariables and 1 is a vector of dimension N+1 with unit elements.Moreover, the k−th element of the vector function E_(ν)=[E_(ν)(0), . . ., E_(ν)(N)]^(T) is defined by

$\begin{matrix}{{E_{V}(k)} = {\max\limits_{0 \leq i \leq k \leq N}{{v\mspace{11mu}(i)}}}} & (19)\end{matrix}$

It is worthwhile to note that in the above proposition, the value of γis an upperbound for the additive model uncertainty with respect to thegiven set of experimental data. Finding an upperbound for the additivemodeling error for all possible experimental data, known in theliterature as the so-called worst-case uncertainty bounding, has beenthe subject of many researches in the past several years, particularlyfor case of LTI uncertainties. However, the problem of computing aworst-case upperbound for non-linear uncertainties still remains an openissue.

Case B1: When Only β is Unknown and Load Dynamics is Stable

FIG. 4 is a block diagram of an uncertainty model structure according toanother embodiment of the present invention, referred to as Case B1. Inparticular, FIG. 4 illustrates a model structure for representation ofuncertainty in bulk modulus coefficient when load dynamics are stable.Ĝ⁻¹ is a model for an integrator.

As mentioned previously, the model structure (14) is a special case forrepresenting the uncertainty in actuator dynamics. To demonstrateanother example, let us consider the case when the only source ofparametric uncertainty is the error in bulk modulus coefficient β. Thiserror indicates that the effect of velocity has not been perfectlyeliminated by the linearizing controller. Applying the linearizingcommand signal u* with nominal {circumflex over (β)} implies that{dot over (τ)}−ν={tilde over (β)}D ² P(x)ω

Now let the mapping from τ+τ_(ext) to ω to w be expressed by ω=

(τ+τ_(ext)) where τ_(ext) represents any external torque and

is an operator representing load dynamics. Define the uncertain block asΔ=D²P(x){tilde over (β)}, then{dot over (τ)}=ν+Δ

τ+Δ

τ_(ext)  (20)

Note that Δ is still a non-linear operator due to the presence of P(x).As it can be seen from FIG. 4, the uncertainty in this case appears in afeedback connection with an integrator. If load dynamics is unknown,

should be contained in Δ. But, in the sequel we assume

is known. In this case the transfer function associated with

(i.e. L(s)) plays the role of a weighting function as W does in (14).Given N+1 samples of ν and τ, the following model structure is then usedfor identification of the model Ĝ, and an upperbound, γ for theuncertainty Δ, where ∥Δ∥₁≦γ:Ĝτ=ν+ΔWτ+e  (21)where e e=ΔLτ_(ext) represents the effect of external disturbance toactuator dynamics resulting from external torque τ_(ext). Obviously,∥e∥_(∞)≦∥Δ∥₁∥L∥₁∥τ_(ext)∥_(∞)=γ∥L∥₁∥τ_(ext)∥_(∞)

Note that here Ĝ stands for the derivative operator (rather thanintegrator operator in model structure (14)). Moreover, we assume that abound on both ∥L∥₁ and ∥τ_(ext)∥_(∞) are known. With the same argumentas in proof of Proposition 1, one can show that the following linearprogramming problem solves the identification problem

${subject}\mspace{14mu}{to}\text{:}\begin{matrix}\underset{\hat{l},\gamma}{\min\;\gamma} \\{{{{T\;\tau\overset{\_}{F}\hat{l}} - v_{N}}} \leq {\gamma{L}_{1}\left( {E_{\tau} + {{\tau_{ext}}_{\infty}\underset{\_}{1}}} \right)}}\end{matrix}$

Case B2: When Only β is Unknown and Load Dynamics are Unstable

Equation (20) can be written as

⁻¹{dot over (τ)}=

⁻¹ν+Δτ+Δτ_(ext)  (22)The following model structure is then proposedĜ₁τ=Ĝ₂ ν+Δτ+e  (23)where Ĝ₁ represents a model for sL(s) and Ĝ₂ is a model for L(s)⁻¹.Obviously, Ĝ₁ and Ĝ₂ include load dynamics and hence, load dynamics isconsidered unknown. Similarly, e represents the term Δτ_(ext) and it isbounded by ∥e∥_(∞)≦∥Δ∥₁∥τ_(ext) ∥_(∞)=γ∥τ_(ext)∥_(∞).

The models Ĝ₁ and Ĝ₂ are parameterized as in (16) with parameter vectors{circumflex over (l)}₁ and {circumflex over (l)}₂, respectively.Therefore, given N+1 samples of ν, τ together with knowledge of∥τ_(ext)∥_(∞), the identification problem is to find {circumflex over(l)}₁ and {circumflex over (l)}₂ and minimum value of γ such that themodel structure (23) is not invalidated by data. The following linearprogramming can be similarly shown to solve the identification problem

${subject}\mspace{14mu}{to}\text{:}\mspace{14mu}\begin{matrix}\begin{matrix}{\min\;\gamma} \\{{{\hat{l}}_{1}.{\hat{l}}_{2}},\gamma}\end{matrix} \\{{{{T_{\tau}F_{1}{\hat{l}}_{1}} - {T_{v}{\overset{\_}{F}}_{2}{\hat{l}}_{2}}}} \leq {\gamma\left( {E_{\tau} + {{\tau_{ext}}_{\infty}\underset{\_}{1}}} \right)}}\end{matrix}$

Remark 1 It is possible that the parametric and non-parametricuncertainties that are represented by Δ are time-varying. In otherwords, if θ varies with time then {tilde over (θ)} will be alsotime-dependent. All previous identification results are still valid inthis case. However, it should be noted that when parameter variation isvery significant (and assuming that the overall identification procedureis long enough to capture the variation of θ), the estimated bound for Δcan be large. This means that the robust external controller designedbased on this large upperbound will be conservative. Apart from anadaptive approach, one way to resolve this problem is to repeat theidentification-controller design in some time intervals. Clearly, thesetime intervals should be long enough to let the identification andcontroller design procedure be completed while, on the other hand, shortenough to be capable of following parameter variations. Such arepetitive identification-robust control design has been used inliterature for slow-varying systems. In our particular case thepreferable minimum time interval turns to be about 2 minutes. However,our implicit assumption is that the actuator operates in steady stateand parameters do not vary significantly.

External Optimal l₁-H_(∞) Controller Design

The nominal model Ĝ together with the uncertainty upperbound γ can beused in a robust control strategy for designing the external lineardiscrete-time controller C that maps the torque error signal {tilde over(τ)} to the new input signal ν. In the sequel, we specify differentrobust stability and performance conditions for the entire closed-loopsystem according to three model structures (14), (21) and (23).

Case A

The nominal output sensitivity function can be defined as S=(1+{tildeover (G)}C)⁻¹ and the nominal input sensitivity function can be definedas S_(u)=CS. The additive uncertain structure (14) induces a robuststability condition on the nominal input sensitivity function∥WS_(u)∥₁<γ⁻¹  (24)

Moreover, in order to attenuate the effect of high frequency sensornoise on the input signal ν, and to limit the amplitude of the inputsignal ν, an H_(∞) constraint should be imposed on the input sensitivityfunction

$\begin{matrix}{{{W_{n}S_{u}}}_{\infty} \leq \delta_{n}} & (25) \\{\min\limits_{C}{{S\mspace{11mu}\left( {\hat{G},C} \right)}}_{1}} & (26)\end{matrix}$

For the feedback loop shown in FIG. 1, the torque tracking error {tildeover (τ)}=τ_(ref)−τ0 and the reference signal τ_(ref) are related to theoutput sensitivity function by {tilde over (τ)}=Sτ_(ref). A presentlydesirable performance objective is to minimize the maximum (over allpossible reference signals) peak-to-peak torque tracking error. This isequivalent to minimize the l⁻¹ norm or the induced I_(∞) norm of theoutput sensitivity function.

Since H₂ and H_(∞) norm of any LTI system are bounded by its l₁ norm,this performance objective obviously minimizes an upperbound for the H₂and H_(∞) norm of the output sensitivity function. This is a propertythat no optimal H₂ or H_(∞) controller possesses. The minimization of∥S∥₁ also minimizes the effect of external disturbance e in τ, howeverone should note that due to the presence of non-linear operator Δ inmodel structure (14), a bounded disturbance can destabilize the systemdepending on initial conditions and nature of non-linearity. Therefore,one should keep in mind that the effect of external disturbance isminimized as long as initial conditions are sufficiently close to systemequilibrium point.

Case B1

Given the model structure (21), the robust stability condition becomes

$\begin{matrix}{{{W\frac{{\hat{G}}^{- 1}}{1 + {{\hat{G}}^{- 1}C}}}} < \gamma^{- 1}} & (27)\end{matrix}$

Recalling that W=L it becomes clear that for the loads with highflexibility, the frequency response of W(jω) is large in some resonantfrequencies. Therefore, constraint (27) requires that the closed loopsensitivity function

$\frac{{\hat{G}}^{- 1}}{1 + {G^{- 1}C}}$be small in load resonance frequencies. This implies that when theeffect of velocity is not perfectly eliminated by the linearizingcontroller, a limitation is imposed on the achievable performancethrough a robust stability constraint. This result is in accordance withknown analysis describing the limitation effects of lightly damped modesof load on the achievable performance of force controllers. It isworthwhile to notice that these lightly damped modes affect theperformance of our proposed controller only when the effect of velocityis not perfectly compensated. However, these modes limit the performanceof typical PID force controllers even in absence of uncertainty inactuator dynamics, due to particular structure of these controllers.

In order to minimize the amplitude of tracking error, the sameperformance objective as in (26) can be considered herein. Note thathere in definition of all sensitivity functions S,S_(u), one shouldreplace Ĝ by Ĝ⁻¹.

Case B2

Similar to case Δ and case B1, model structure (23), induces a robuststability constraint such as

${\frac{1}{{\hat{G}}_{1} + {{\hat{G}}_{2}C}}}_{\infty} < \gamma^{- 1}$on the external controller C. Note that since {tilde over (G)}₁represents sL(s) and L(s) is unstable, the transfer function

$\frac{{\hat{G}}^{- 1}}{1 + {G^{- 1}C}}$is non-minimum phase and this fact can impose some limitations on theachievable performance of torque controller.

Limit-Cycle

When the gain of external controller is high, a self excitingoscillation (limit-cycle) is observed in the generated torque. FIG. 7graphically illustrates a limit cycle in actuator dynamics in thepresence of a high gain external controller. According to experimentalresults, the frequency of the main harmonic of the limit-cycle can rangefrom 200 to 400 rad/sec depending on controller gain. Moreover, thefrequency is almost independent of load inertia. As seen in FIG. 7, theoscillations start when velocity approaches zero, suggesting that theoscillations may be caused by static friction. The valve dead-zone canbe also a cause because its effect is dominant near zero velocity thatcorresponds to low flow and small valve opening. Moreover, theoccurrence of limit-cycle depends also on reference torque. For example,for high frequency sinusoidal reference signals (higher than 1 Hz), theoscillations do not normally occur. The existence of limit-cyclephenomenon has been reported and it has been attributed to the existenceof electromagnetic hysteresis in the valve dynamics. It has been shownthat the oscillations can be caused by lightly damped modes of loaddynamics.

If limit-cycle is considered to be caused only by load dynamics, itseffect is injected into torque dynamics (4) through the velocity signalω. The effect of velocity can then be perfectly compensated by thelinearizing controller in absence of uncertainty and especially inabsence of parametric error in bulk modulus β. However, when {tilde over(β)} is not zero, a constraint like (27) should be imposed to ensurerobust stability with an acceptable level of oscillation attenuation. Onthe other hand, if limit-cycle originates from actuator dynamics, itseffect can be represented by an uncertainty term as in model structure(14). A constraint like (24) can then ensure the stability robustnesstogether with a level of performance. Another efficient way to attenuatethe oscillations is to impose point-wise constraints on the inputsensitivity function S_(u) (as defined in case A) in the frequencyranges where the oscillations occur. These constraints aim to cancel outthe effect of the main harmonics of the limit-cycle by preventing themto be injected into the system through ν. We describe these constraintsby|S _(u)(jω _(κ))|≦δ_(κ), for κ=1, . . . m  (28)

In any case, the existence of limit-cycle obviously limits theachievable performance of the torque controller.

Synthesis

From a synthesis point of view, there are more constraints that shouldde imposed on the output sensitivity function S. For example, if thenominal model Ĝ has any unstable pole-zeros or pure delays, thecomplementary sensitivity function 1-S should also contain exactly thesame dynamics in order to avoid any unstable pole-zero simplificationbetween the controller and the model. These constraints are usuallyreferred to as zero interpolation conditions, and they are transformedinto LP constraints.

A synthesis procedure according to an embodiment of the presentinvention for designing the external controller C is based on theformulation of convex Linear Matrix Inequality (LMI) or LinearProgramming (LP) constraints for each of the control designspecifications (24)-(26). In general, these constraints areinfinite-dimensional but in many cases they can be reduced tofinite-dimensional optimization. For example, it is known that in SISOcase, a pure l minimization in (26) has an Finite Impulse Responsesolution for S. Moreover, the minimization problem (26) together with(28) has typically an FIR solution for S. However, by imposing all thecontrol specifications (24)-(25), the optimal solution for S can be nolonger FIR. One way to check this property is to approximate allinfinite-dimensional constraints by finite-dimensional ones via finitelymany variables and finitely many equations methods. Here in our problem,we considered an FIR structure for the output sensitivity function S.The interpolation conditions as well as the control specifications (24)and (26) are consequently transformed into LP constraints. Furthermore,by the application of Bounded Real Lemma, the H_(∞) constraint (25) istransformed into an LMI constraint. Also, the point-wise constraints in(28) are transformed into an LMI by using methods known in the art.

Experimental Results

The experimental tests have been conducted on the pitch actuator of theTitan II Schilling industrial robot which is located at the roboticslaboratory of the Canadian Space Agency. In a particular experimentalresult, the joint is driven by a vane type rotary hydraulic actuatorthat generates a nominal torque of 500 Nm at nominal supply pressure of3000 Psi. The position angle of the actuator can vary between −90° to+90° and it is measured by a 16-bit encoder. The maximum velocity of theactuator is 192°/sec. The chamber pressure p₁ and p₂ are measured by twopressure transducers. All analog signals are sampled at 1 kHz.

Identification and Uncertainty Bounding

The parameters of the actuator non-linear model are estimated via atypical least-squares optimization (10) using 1000 time-domain datasamples. The identified parameters ĉ_(p), ĉ₁+ĉ_(el) and {circumflex over(β)} of the non-linear model are shown in Table 1.

TABLE 1 V₀ D ĉ_(p) ĉ_(l) + ĉ_(el) {circumflex over (β)} 1.67 × 10⁻⁴ 2.66× 10⁻⁵ 1.49 × 10⁻⁴ 1.11 × 10⁻⁷ 1.44 × 10⁴

The identification procedure typically needs computation of {dot over(τ)} through numerical differentiation. In order to decrease noiseamplification during differentiation, we decimated τ with a factor of 5before differentiation. As discussed earlier, the choice of input signalu has a strong impact on the condition number of the regression matrix γwhich in its turn can affect the parametric error. It is known that forpersistently exciting input signals with wide frequency bandwidth, thiscondition number is close to one and consequently the parametric erroris reduced. However, when such persistent input signals are applied in ahydraulic actuator they can cause sharp variation of torque signal τ,which can complicate the numerical differentiation of τ needed foridentification purpose. So it seems that there is a compromise betweenthe degree of persistency of input signal and the degree of difficultyin numerical differentiation of τ.

After identification of actuator parameters and in order to validate thenon-linear model, we computed the estimated input signal û from equation(4) using the identified parameters and the measured signals {dot over(τ)},p₁,p₂,ω and x. The estimated input û was compared with the measuredinput signal u. FIG. 5 graphically illustrates a validation of anon-linear model according to an embodiment of the present invention,which shows a satisfactory match between u and û in low frequencies.Taking u as input and τ as output to actuator dynamics, this comparisonis in fact a measure of matching between the true and the identifiedinverse non-linear dynamics of the actuator.

Ideally, the identified parameters, which are used to compute thefeedback linearization control law (8), results in an integrator systemmapping the new input ν to actuator torque τ. In practice, however, themapping deviates from an integrator due to unmodeled dynamics. Usingmodel structure (14) we identified the nominal model Ĝ, assuming W=1,

${\hat{G}\left( q^{- 1} \right)} = {q^{- 2}\frac{0.1313 - {0.0603\; q^{- 1}}}{1 - {0.994\; q^{- 1}}}}$via the identification procedure described earlier. The frequencyresponse of Ĝ is shown in FIG. 6, graphically illustrates a frequencyresponse of a linearized model represented by an uncertainty modelstructure according to an embodiment of the present invention. It isevident from the figure that Ĝ behaves as an integrator within frequencyrange of 0.17 and 170 Hz. For the given data, the upperbound of theadditive uncertainty is calculated to be ∥Δ∥₁≦0.16.

Optimal l₁-H_(∞) Robust Control Design

The synthesis of C is based on models structure (14) and constraintspresented in section 5.1. Since the l₁ norm of the additive non-linearuncertainty is bounded by 0.16, we must have ∥S_(u)∥₁<0.16⁻¹=6.25 tomaintain robust stability. Moreover, in order to attenuate the effect ofthe noise on the new input signal ν, we specify high pass filter W_(n)as weighting function in constraint (25)

${W_{n}\left( q^{- 1} \right)} = \frac{1 - {0.99\; q^{- 1}}}{5.22 - {4.23\; q^{- 1}}}$Note that in SISO case the constraint (25) is equivalent to bound|S_(u)(jω)|by|W_(n) ⁻¹(jω)|

As discussed previously, the system may exhibit limit-cycle if high gainlinear controller is used. FIG. 7 shows this phenomena when the linearcontroller is simply a proportional gain (C=K=2). The frequency range ofthe principal harmonic of the limit-cycle for different values of K isbetween 200 and 400 rad/sec. In order to attenuate the limit-cycleoscillation, we imposed two point-wise constraints on the inputsensitivity function in 200 and 400 rad/sec. with δ₁=δ₂=6 dB, as statedin (28).

The controller synthesized based on all these design specifications is a21th order discrete-time transfer function. The optimal controller givesan output sensitivity function with ∥S∥₁=2.04 which implies that theamplitude of tracking error for any reference signal with∥τ_(ref)∥_(∞)≦1 does not exceed 2.04.

Note that the complexity of external controller is a natural consequenceof imposing several robust stability and performance constraints. Forexample, in case of pointwise constraints (28), the controller needs toinclude narrow-band behavior in two different frequencies andconsequently its order becomes high. Moreover, unlike H_(∞) control, inl₁ case, no direct relationship exists between the order of optimalcontroller and that of the model. This means that the order of optimall₁ controllers can be arbitrarily high regardless of model order.However, in H_(∞) or H₂ case, the controller order is bounded by orderof model and weighting functions.

From implementation point of view, since the resulting controller isdesigned in discrete-time domain, its implementation does not need anycontinuous-to-discrete transformation. Although in none of ourexperiments we had an implementation problem due to complexity of C,standard model-reduction techniques can be applied provided that thereduced order controller does not violate key stability and performancecriteria.

Performance Evaluation

In order to demonstrate back-drivability (equivalently low sensitivityof the controlled actuator to velocity and external torqueperturbations), we conducted an experiment in presence of two types ofcontrollers. FIG. 8 graphically illustrates frequency responses ofresulting input and output sensitivity functions together with weighingfunctions. FIG. 9 graphically illustrates an effect of external torquedisturbance on generated torque, specifically the response of thehydraulic torque to external torque disturbance with and without havinginner feedback linearization loop, respectively.

During the experiment, the end-effector of the robot was moved by handwhile controllers were regulating the torque of pitch actuator to zero.It is evident from the figures that the sensitivity of the controlsystem to external torque disturbance is substantially reduced when thefeedback linearization is used. Let us define backdrivability index ofan actuator as the ratio of torque amplitude to velocity amplitude whenτ_(ref)≡0 and when actuator is subject to external torques

$\eta = \left. \frac{{\tau }_{\infty}}{{\omega }_{\infty}} \right|_{{{\tau\;{ref}} = 0},{{\tau\;{ext}} \neq 0}}$

Obviously, for an ideal source of torque η=0. This index is also ameasure of impedance of actuator. Without feedback linearization innerloop (a), the backdrivability index is 27 but for the proposed cascadecontroller (b), this quantity decreases to 7.9. Low sensitivity toexternal torque disturbance in case (b) implies that the hydraulicsystem is backdrivable and performance of torque controller is not muchaffected by load variations or external torques.

FIG. 10 shows the step response of the proposed controller, alsodescribed as step tracking of actuator torque in the presence ofvelocity. The rise-times and the settling times of the control system aswell as sensitivity to velocity disturbance are reported in Table 2:

TABLE 2 t_(r) ⁺ t_(r) ⁻ t_(s) ⁺ t_(s) ⁻ η 22 ms 21 ms 200 ms 100 ms 7.9

FIGS. 11, 12 and 13 show the sinusoidal reference tracking for differentfrequencies and in presence of velocity disturbance. Specifically, thefigures illustrate 0.5 Hz, 2 Hz, and 5 Hz sine tracking, respectively,of actuator torque in presence of velocity. Note that because of thesmoothness of sinusoids, the limit-cycle phenomenon might not beobserved for high-gain controllers.

In many robotics applications such as contact force control, thehydraulic actuators motion is negligible. Therefore, the distortioncaused by velocity in the generating torque is reduced. In order tocheck the performance of the designed controller in absence of velocity,we conducted the previous experiments when robot end-effector waslocked.

Comparison of Step Response of System

In presence and in absence of velocity in FIG. 10 and FIG. 14, shows aslight improvement in performance of controller in absence of velocity.FIG. 14 graphically illustrates step response of actuator torque inabsence of velocity, while FIGS. 15-17 graphically illustrate 0.5 Hz, 2Hz, and 5 Hz sine tracking, respectively, of actuator torque in absenceof velocity. Obviously, if the effect of velocity is well compensated bythe linearizing controller one should not expect a significantdifference between the performance of composite controller in static andnon-static case. The tracking performance of controller with respect tosinusoidal reference inputs with and without presence of velocitydisturbance are illustrated in FIGS. 11, 12, 13 and FIGS. 15, 16, 17,respectively.

Conclusion

A novel combined scheme for identification and robust torque control ofrotary hydraulic actuators has been presented. The feedbacklinearization loop has been used to linearize the actuator dynamics andto compensate for non-linear effect of velocity disturbance, while theouter l₁-H_(∞) optimally loop has been implemented to ensure best degreeof achievable robustness and performance for the system in the face ofpossible uncompensated non-linearities. The stability analysis of theinternal dynamics provides some necessary results that could beconsidered in developing new methods for design of the torquecontrollers achieving global stability. The experimental resultsdescribed herein illustrate the implementability and efficiency of theproposed combined scheme.

As will be understood by those of skill in the art, the methods ofdesign, methods and systems relating to torque/force controllerembodiments of the present invention can be generally embodied assoftware residing on a general purpose, or other suitable, computerhaving a modem or internet connection to a desired optical network. Theapplication software embodying methods of design, methods and/or systemsrelating to torque/force controller embodiments of the present inventioncan be provided on any suitable computer-useable medium for execution bythe computer, such as CD-ROM, hard disk, read-only memory, or randomaccess memory, or as part of any carrier signal or carrier wave. In apresently preferred embodiment, the application software is written in asuitable programming language, such as C++ or Matlab/Simulink, and isorganized, as described above, into a plurality of modules or elementsthat perform the method steps described. The methods could beimplemented in a digital signal processor (DSP) or other similarhardware-related implementation. When reference is made to a means forperforming steps of methods according to an embodiment of the presentinvention, such means are to be understood to include computer-readablemeans as described above.

Therefore, according to an aspect as described above, the presentinvention provides a computer-readable storage medium, comprisingcomputer instructions for: calculating a linearizing control law basedon identified parameters of an actuator non-linear model; determining anuncertainty model for the actuator, the uncertainty model including anon-linear component; estimating an uncertainty bound based on theidentified parameters; and designing an external linear controller basedon the calculated linearizing control law, the determined uncertaintymodel, and the estimated uncertainty bound.

Although embodiments of the present invention have primarily beendescribed in conjunction with a torque/force controller, it is to beunderstood that such a controller can be provided separately, or can beprovided integrally with an actuator. For instance, a manufacturer orreseller of hydraulic actuators could include a torque/force controlleraccording to an embodiment of the present invention integral with thehydraulic actuator. As such, in an aspect, there is provided a hydraulicactuator, comprising a controller according to embodiments of thepresent invention. In another aspect, there is provided a hydraulicactuator comprising a computer-readable storage medium, comprisingstatements and instructions which, when executed, cause a computer (or aprocessor in data communication with the actuator) to perform a methodaccording to embodiments of the present invention.

Embodiments of the present invention can find application in associationwith hydraulic actuators used in many applications. For instance, theyare widely used to drive robotic manipulators in industry for earthmoving, material handling, and in the areas of construction, forestryand manufacturing automation. Other applications include high powerindustrial machinery such as machine tools, aircraft, material handling,construction, mining, and agricultural equipment. Robotic uses includeassembly tasks, mobile robots, and robotic applications in space, forexample with Special Purpose Dexterous Manipulators (SPDMs) such as theCanadarm™. General engineering applications include vibration isolationand automobile active suspension. Military applications also exist inaerospace, aviation, submarines and maritime applications.

The above-described embodiments of the present invention are intended tobe examples only. Alterations, modifications and variations may beeffected to the particular embodiments by those of skill in the artwithout departing from the scope of the invention, which is definedsolely by the claims appended hereto.

1. A composite controller for a hydraulic actuator, the hydraulicactuator for generating a manipulating influence to be applied to aload, the controller comprising: a dynamic feedback linearizingcontroller inner loop for controlling the hydraulic actuator inaccordance with a stored linear model representing non-linear dynamicbehaviour of an unloaded hydraulic actuator; and a robust linearcontroller outer loop for compensating for non-linearities in the linearmodel based on an uncertainty model representing deviation of the linearmodel from linearity, the uncertainty model having a non-linearcomponent and a linear time invariant component and an estimated upperbound, the robust linear controller having time invariant dynamics. 2.The controller of claim 1 wherein the inner loop and the outer loopco-operate to permit the hydraulic actuator to generate a desiredmanipulating influence irrespective of motion of the hydraulic actuator.3. The controller of claim 1 wherein the hydraulic actuator includes ajoint, and wherein the non-linear dynamic behaviour of the unloadedhydraulic actuator is obtained by substantially minimizing effects ofthe load on the manipulating influence by perturbing the linear model inresponse to a velocity of the joint.
 4. The controller of claim 1wherein the linear model is based on measured linear parameters of thehydraulic actuator.
 5. The controller of claim 1 wherein the linearizingcontroller includes means for obtaining the linear model based on alinearizing control law for the hydraulic actuator.
 6. The controller ofclaim 5 wherein the linearizing controller includes means fordetermining the linearizing control law for the hydraulic actuator. 7.The controller of claim 1 further comprising means for calculating theestimated upper bound.
 8. The controller of claim 1 wherein themanipulating influence comprises a torque.
 9. The controller of claim 1wherein the manipulating influence comprises a force.
 10. The controllerof claim 1 wherein the hydraulic actuator is a rotary hydraulicactuator.
 11. The controller of claim 1 wherein the hydraulic actuatoris a linear hydraulic actuator.
 12. A method of designing a hydraulicactuator controller, comprising: determining an uncertainty model tocompensate for differences between a linearized model of the hydraulicactuator and actual behaviour of the hydraulic actuator, the uncertaintymodel including a non-linear component and a linear time invariant modelcomponent; estimating an uncertainty upper bound, for the uncertaintymodel, based on identified parameters of non-linear behaviour of theactuator; and designing a robust linear controller based on thedetermined uncertainty model and the estimated uncertainty upper boundso that the robust linear controller has time invariant dynamics. 13.The method of claim 12 wherein the non-linear component is based onunmodeled actuator dynamics.
 14. The method of claim 12 wherein the stepof estimating the uncertainty upper bound comprises: applying a dynamicfeedback linearizing inner loop controller in conjunction with therobust linear controller in an outer loop; exciting the robust linearcontroller and obtaining measured values of input and output signals;and determining, based on the measured values of input and outputsignals, a valid value of the uncertainty upper bound the uncertaintymodel.
 15. The method of claim 12 wherein the step of designing therobust linear controller comprises imposing robust stability andperformance constraints based on characteristics of the uncertaintymodel.
 16. The method of claim 12 further comprising the step ofcalculating a linearizing control law based on the identified parametersof non-linear behaviour of the actuator.
 17. The method of claim 16wherein the linearizing control law comprises a dynamic feedbacklinearizing control law.
 18. The method of claim 12 further comprisingthe step of extracting the identified parameters based on measuredsignals.
 19. A computer-readable storage medium, comprising statementsand instructions which, when executed, cause a computer to perform themethod of claim 12.